Asymptotic behavior of some factorizations of random words

In this paper, we consider the normalized lengths of the factors of some factorizations of random words. First, for the Lyndon factorization of finite random words with n independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution, we prove that the limit law of the normalized lengths of the smallest Lyndon factors is a variant of the stickbreaking process. Convergence of the distribution of the lengths of the longest factors to a Poisson–Dirichlet distribution follows. Second, we consider the standard factorization of random Lyndon word : we prove that the distribution of the normalized length of the standard right factor of a random n ‐letters long Lyndon word, derived from such an alphabet, converges, when n is large, to:μ ( d x ) = p 1δ 1 ( d x ) + ( 1 − p 1 ) 1 [ 0 , 1 ) ( x ) d x , $$ \mu (dx)={p}_1{\delta}_1(dx)+\left(1-{p}_1\right){\mathbf{1}}_{\left[0,1\right)}(x) dx, $$ in whichp 1$$ {p}_1 $$ denotes the probability of the smallest letter of the alphabet.